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I am wondering which is the earliest record of the fact that the plane can be tiled by regular hexagons (in addition to triangles and squares, which may be slightly more obvious).

Had a look in the Loeb volume Greek Mathematics from Thales to Euclid, but could not find it (they discuss hexagons but not tiling the plane).

Kepler, Harmony of the World (1619) is the earliest I have found so far.

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It depends somewhat on what qualifies as a "record". People observed honeycombs in bee-hives since the prehistoric times, so any evidence of honey consumption would be a record of sorts. Aristotle mentions honeycombs explicitly in his History of Animals, for example.

As for a theoretical geometric study of tesselations, it occurs surprisingly late. The earliest surviving one is indeed Kepler's of 1619, unless we count Platonic and Archimedean solids as tesselations of the sphere. It is interesting that (presumably) Theaetetus's angle counting argument in the Elements for the existence of five Platonic solids is very similar to the standard modern argument for the existence of three plane tesselations. But we can only speculate whether he took any interest in those. Some Roman mosaics display hexagonal tiling patterns, especially those themed on Orpheus, see e.g. Roman Mosaics in Getty Museum.

There is more indirect evidence from medieval Islamic art. Lu and Broug even claim in Classifying Hexagonal Tilings in Islamic Architecture that:

"The drafting of a hexagonal honeycomb grid using a compass and straightedge was well understood both by the ancient Greeks and by medieval Islamic architects and designers. The specific sequence of steps using the compass and straightedge to create the honeycomb tessellation of hexagons has been well illustrated in a number of publications."

However, this seems to be an indirect inference. They give no examples from antiquity, and the earliest Islamic examples are the Dado panel in Nishapur, Iran from the 10th century AD (currently kept in the Metropolitan Museum of Art in New York), and the Great Mosque of Cordoba, Spain (987 AD). Abdullahi and Bin Embi confirm the timeline in Evolution of Islamic Geometric Patterns:

"These developments began with simple geometrical shapes constructed from a circle and a set of tangential circles with the same radius (Critchlow, 1976), as seen in the Great Mosque of Kairouan during the early 9th century. By the late 9th century, grids of circles were introduced in the Ibn-Tulun Mosque. The grids were used as constructive bases for the simplest regular and semi-regular tiling with equilateral triangles, squares, hexagons, and octagons."

Then, contemporary to rise of Persian philosophers and cosmologists from Abu Sahl Al-Tustari to Sohravadi who had debates and important contribution to nature of numbers and their relation with that of nature (Critchlow, 1989), mystical Tetractys motifs and symbol merged to traditional geometric patterns. The result was the invention of abstract 6-point geometrical patterns based on the Tetractys symbol and 12-point star patterns that are associated with 12 zodiacal sectors. These decorative elements adorn the facades of the Tomb of Kharaqan (1067) in Iran."

Abu al-Wafa' Buzjani's (959-998 AD) Geometry Needed by Craftsmen also dates to this period. But it is unclear that the early girih designs were conceptualized as tilings, the dominant early method was "direct strapwork" of zigzagging lines using straightedge and compass. Lu and Steinhardt argue that the shift occurred later:

"On the basis of our examination of a large number of girih patterns decorating medieval Islamic buildings, architectural scrolls, and other forms of medieval Islamic art, we suggest that by 1200 C.E. there was an important breakthrough in Islamic mathematics and design: the discovery of an entirely new way to conceptualize and construct girih line patterns as decorated tessellations using a set of five tile types, which we call “girih tiles”... We further show how the girih-tile approach opened the path to creating new types of extraordinarily complex patterns, including a nearly perfect quasi-crystalline Penrose pattern on the Darb-i Imam shrine (Isfahan, Iran, 1453 C.E.), whose underlying mathematics were not understood for another five centuries in the West."

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    $\begingroup$ That hexagons cover the plane has been known as an empirical fact. Varro Res Rusticae III.16 comments on the hexagon that bees use for their honeycomb. Proving obvious things is a Greek pass time, quarreling what is a proof is a modern one. Exercises: prove that there are no 7 legged horses; prove that except for 3,4 and,6 no other regular polygon covers the plane. $\endgroup$ – sand1 Jun 1 '19 at 12:55
  • $\begingroup$ @TomasBy Honeycombs are essentially two-dimensional, just cylinders over a plane tesselation, the visible cross-sections are hexagonal tilings. $\endgroup$ – Conifold Jun 1 '19 at 17:41
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Ornaments corresponding to all possible discrete groups of motions acting on the plane can be found in the wall decoration of Alhambra palace in Granada (Spain). This was built in 13-15 centuries (it was rebuilt several times, and it is not exactly known who and when made these decorations). It is amazing that they knew all these groups.

This gives an upper estimate for the date you ask. I am sure that the hexagonal tiling was known long before, but this knowledge is hard to confirm with documents and artifacts.

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  • $\begingroup$ Yes, I'm sure the Ancients knew it, but I want something I can cite. $\endgroup$ – Tomas By Jun 1 '19 at 0:54
  • $\begingroup$ I suppose there is nothing you can cite about the ancients. As far as I know, no depiction or description survived. $\endgroup$ – Alexandre Eremenko Jun 1 '19 at 1:02
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    $\begingroup$ Honeycombs. Also, hexagons are seen in a triangular mesh. Looking at ancient mosaics would evidence a date when people knew this, at least practically. Discussions of Euclide's proof about the 5 regular solids would include a mention, or perhaps Archimedes does it. $\endgroup$ – sand1 Jun 1 '19 at 8:32
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    $\begingroup$ @sand1 yeah but bees never documented a proof :-) $\endgroup$ – Carl Witthoft Jun 3 '19 at 13:18
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    $\begingroup$ It is not universally accepted that whoever made the Alhambra tilings knew about these patterns (they certainly didn't know about groups). See for example Branko Grunbaum - What symmetry groups are present in the Alhambra $\endgroup$ – Jishin Noben Jun 4 '19 at 7:38

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